Rigorous Results for Hierarchical Models of Structural Glasses

2014/03/31 by Michele Castellana
Computer Science · Engineering · Mathematics · Physics and Astronomy · #3D Shape Modeling and Analysis #Computer Graphics and Visualization Techniques #Computer science #Econometrics #Geology #Mathematics #Physics #Statistical physics #Theoretical and Computational Physics #cond-mat.dis-nn #math-ph #math.MP

paper · pdf · doi:10.1007/s10955-014-1085-9

published as Journal of Statistical Physics 157(2), 219 (2014)

openalex publication_date 2014/08/13 · arxiv created 2014/08/26 · arxiv updated 2014/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We consider two non-mean-field models of structural glasses built on a hierarchical lattice. First, we consider a hierarchical version of the random energy model (HREM), and we prove the existence of the thermodynamic limit and self-averaging of the free energy. Furthermore, we prove that the infinite-volume entropy is positive in a high-temperature region bounded from below, thus providing an upper bound on the Kauzmann critical temperature. In addition, we show how to improve this bound by leveraging the hierarchical structure of the model. Finally, we introduce a hierarchical version of the p-spin model of a structural glass, and we prove the existence of the thermodynamic limit and self-averaging of the free energy.

Citations

• The Random First-Order Transition Theory of Glasses: a critical assessment
• Hierarchical Random Energy Model of a Spin Glass
• The Thermodynamic Limit in Mean Field Spin Glass Models
• Through the Glass Lightly
• The Nature of the Glassy State and the Behavior of Liquids at Low Temperatures.
• Free-Energy Bounds for Hierarchical Spin Models
• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model